NAG Library Routine Document

G05KKF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G05KKF allows for the generation of multiple, independent, sequences of pseudorandom numbers using the skip-ahead method. The base pseudorandom number sequence defined by STATE is advanced

2^{n}

places.

2 Specification

SUBROUTINE G05KKF (

N, STATE, IFAIL)

INTEGER

N, STATE(*), IFAIL

3 Description

G05KKF adjusts a base generator to allow multiple, independent, sequences of pseudorandom numbers to be generated via the skip-ahead method (see the G05 Chapter Introduction for details).

If, prior to calling G05KKF the base generator defined by STATE would produce random numbers

x_{1}, x_{2}, x_{3}, \dots

, then after calling G05KKF the generator will produce random numbers

x_{2^{n} + 1}, x_{2^{n} + 2}, x_{2^{n} + 3}, \dots

One of the initialization routines G05KFF (for a repeatable sequence if computed sequentially) or G05KGF (for a non-repeatable sequence) must be called prior to the first call to G05KKF.

The skip-ahead algorithm can be used in conjunction with any of the six base generators discussed in the G05 Chapter Introduction.

4 References

Haramoto H, Matsumoto M, Nishimura T, Panneton F and L'Ecuyer P (2008) Efficient jump ahead for F2-linear random number generators INFORMS J. on Computing 20(3) 385–390

Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

5 Parameters

1: N – INTEGERInput: On entry: $n$ , where the number of places to skip-ahead is defined as $2^{n}$ .
Constraint: $N \geq 0$ .
2: STATE( $*$ ) – INTEGER arrayCommunication Array: Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.
On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.
3: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,

N < 0

$IFAIL = 2$

On entry,

STATE vector was not initialized or has been corrupted.

$IFAIL = 3$: On entry, cannot use the skip-ahead method with the base generator defined by STATE.

$IFAIL = 4$: On entry, the base generator is Mersenne Twister, but the STATE vector defined on initialization is not large enough to perform a skip-ahead. See the initialization routine G05KFF or G05KGF.

7 Accuracy

Not applicable.

8 Further Comments

Calling G05KKF and then generating a series of uniform values using G05SAF is equivalent to, but more efficient than, calling G05SAF and discarding the first

2^{n}

values. This may not be the case for distributions other than the uniform, as some distributional generators require more than one uniform variate to generate a single draw from the required distribution.

9 Example

This example initializes a base generator using G05KFF and then uses G05KKF to advance the sequence

2^{17}

places before generating five variates from a uniform distribution using G05SAF.

NAG Library Routine DocumentG05KKF